Actium War Rig is incredibly versatile as it is. Sweet Business doesn't really do anything special other than it's got a huge mag capacity, and it isn't the only auto rifle that benefits from Actium.
Pairing Actium War Rig with the following would make the following OP in certain situations.
Suros Regime - longer Bonus damage and continuous healing.
Monte Carlo - constant melee regen.
Thunderlord - continuous lightning strikes and pretty much melt any overload champion with ease.
Xenophage - longer sustained DPS on bosses, easier one phase on some raid bosses. What are champions?
Heir Apparent - literally never reload.
Grand Overture - easier to get more missiles for Omega Strike for higher DPS.
And those are just a few.
Actium has just the right amount of hell yeah for any job.
English
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Yes, many auto rifles get benefits from actium war rig as it is but those benefits aren't good enough to justify using it over many other exotics. 600 and 450 rpm autos reload fast enough to not use actium in many cases and have decent handling. 360 and 720 rpm autos would benefit from my suggestion the most. Their handling is usually poor with long reloads, here my suggestion would make those archetypes simply more usable. Suros regime is an outlier when using actium, and ups their benefits. At least in spinning up form, focused fire behaves the same as any 360 auto but with 600 auto stats. Same could be said about tommy's matchbook because it allows its 100% damage buff to last longer before reloading. Though the increased reload speed I suggested would benefit the weapon significantly. Machine guns like xenophage and grand overture already allow good dps with actium because you can dump all your ammo without ever reloading, so here my suggestion would do literally nothing. Here actium does show its strength as it is. Thunderlord can do 90-95% of the same damage without actium war rig, at least against single targets. Any legendary machine gun would only benefit from the suggested reload speed buff, increasing their dps over longer period of time. Sweet business can be used without actium war rig but they certainly work better together. Alone, sweet business has relatively long wind up time to its max rpm, very long reload, huge sound in pvp, mediocre damage in harder content (especially because ideally you would want to use it for longer periods of time) and it sacrifises your moblity for that mediocre damage. I call its damage mediocre because while you can kill and shred through red bars, the second you face a shield or tougher enemies you are forced to switch weapons. In -15 plus content atleast. Adaptive munitions or target lock would greatly benefit it in these difficulties and especially when paired with actium war rig could provide enough add killing capability to warrant using in these difficulties.